Efficiency of Truthful and Symmetric Mechanisms in One-sided Matching

We study the efficiency (in terms of social welfare) of truthful and symmetric mechanisms in one-sided matching problems with {\em dichotomous preferences} and {\em normalized von Neumann-Morgenstern preferences}. We are particularly interested in the well-known {\em Random Serial Dictatorship} mechanism. For dichotomous preferences, we first show that truthful, symmetric and optimal mechanisms exist if intractable mechanisms are allowed. We then provide a connection to online bipartite matching. Using this connection, it is possible to design truthful, symmetric and tractable mechanisms that extract 0.69 of the maximum social welfare, which works under assumption that agents are not adversarial. Without this assumption, we show that Random Serial Dictatorship always returns an assignment in which the expected social welfare is at least a third of the maximum social welfare. For normalized von Neumann-Morgenstern preferences, we show that Random Serial Dictatorship always returns an assignment in which the expected social welfare is at least \frac{1}{e}\frac{\nu(\opt)^2}{n}, where \nu(\opt) is the maximum social welfare and $n$ is the number of both agents and items. On the hardness side, we show that no truthful mechanism can achieve a social welfare better than \frac{\nu(\opt)^2}{n}.Comment: 13 pages, 1 figur

Eulerian and modified Lagrangian approaches to multi-dimensional condensation and collection

Turbulence is argued to play a crucial role in cloud droplet growth. The combined problem of turbulence and cloud droplet growth is numerically challenging. Here, an Eulerian scheme based on the Smoluchowski equation is compared with two Lagrangian superparticle (or su- perdroplet) schemes in the presence of condensation and collection. The growth processes are studied either separately or in combination using either two-dimensional turbulence, a steady flow, or just gravitational acceleration without gas flow. Good agreement between the differ- ent schemes for the time evolution of the size spectra is observed in the presence of gravity or turbulence. Higher moments of the size spectra are found to be a useful tool to characterize the growth of the largest drops through collection. Remarkably, the tails of the size spectra are reasonably well described by a gamma distribution in cases with gravity or turbulence. The Lagrangian schemes are generally found to be superior over the Eulerian one in terms of computational performance. However, it is shown that the use of interpolation schemes such as the cloud-in-cell algorithm is detrimental in connection with superparticle or superdroplet approaches. Furthermore, the use of symmetric over asymmetric collection schemes is shown to reduce the amount of scatter in the results.Comment: 36 pages, 17 figure

Social welfare in one-sided matchings: Random priority and beyond

We study the problem of approximate social welfare maximization (without money) in one-sided matching problems when agents have unrestricted cardinal preferences over a finite set of items. Random priority is a very well-known truthful-in-expectation mechanism for the problem. We prove that the approximation ratio of random priority is Theta(n^{-1/2}) while no truthful-in-expectation mechanism can achieve an approximation ratio better than O(n^{-1/2}), where n is the number of agents and items. Furthermore, we prove that the approximation ratio of all ordinal (not necessarily truthful-in-expectation) mechanisms is upper bounded by O(n^{-1/2}), indicating that random priority is asymptotically the best truthful-in-expectation mechanism and the best ordinal mechanism for the problem.Comment: 13 page

Extending the range of error estimates for radial approximation in Euclidean space and on spheres

We adapt Schaback's error doubling trick [R. Schaback. Improved error bounds for scattered data interpolation by radial basis functions. Math. Comp., 68(225):201--216, 1999.] to give error estimates for radial interpolation of functions with smoothness lying (in some sense) between that of the usual native space and the subspace with double the smoothness. We do this for both bounded subsets of R^d and spheres. As a step on the way to our ultimate goal we also show convergence of pseudoderivatives of the interpolation error.Comment: 10 page

Quasi-thermal Comptonization and gamma-ray bursts

Quasi-thermal Comptonization in internal shocks formed between relativistic shells can account for the high energy emission of gamma-ray bursts. This is in fact the dominant cooling mechanism if the typical energy of the emitting particles is achieved either through the balance between heating and cooling or as a result of electron-positron pair production. Both processes yield sub or mildly relativistic energies. In this case the synchrotron spectrum is self-absorbed, providing the seed soft photons for the Comptonization process, whose spectrum is flat [F(v) ~ const], ending either in an exponential cutoff or a Wien peak, depending on the scattering optical depth of the emitting particles. Self-consistent particle energy and optical depth are estimated and found in agreement with the observed spectra.Comment: 10 pages, ApJ Letters, accepted for publicatio

Effect of turbulence on collisional growth of cloud droplets

We investigate the effect of turbulence on the collisional growth of um-sized droplets through high- resolution numerical simulations with well resolved Kolmogorov scales, assuming a collision and coalescence efficiency of unity. The droplet dynamics and collisions are approximated using a superparticle approach. In the absence of gravity, we show that the time evolution of the shape of the droplet-size distribution due to turbulence-induced collisions depends strongly on the turbulent energy-dissipation rate, but only weakly on the Reynolds number. This can be explained through the energy dissipation rate dependence of the mean collision rate described by the Saffman-Turner collision model. Consistent with the Saffman-Turner collision model and its extensions, the collision rate increases as the square root of the energy dissipation rate even when coalescence is invoked. The size distribution exhibits power law behavior with a slope of -3.7 between a maximum at approximately 10 um up to about 40 um. When gravity is invoked, turbulence is found to dominate the time evolution of an initially monodisperse droplet distribution at early times. At later times, however, gravity takes over and dominates the collisional growth. We find that the formation of large droplets is very sensitive to the turbulent energy dissipation rate. This is due to the fact that turbulence enhances the collisional growth between similar sized droplets at the early stage of raindrop formation. The mean collision rate grows exponentially, which is consistent with the theoretical prediction of the continuous collisional growth even when turbulence-generated collisions are invoked. This consistency only reflects the mean effect of turbulence on collisional growth

Social Welfare in One-sided Matching Markets without Money

We study social welfare in one-sided matching markets where the goal is to efficiently allocate n items to n agents that each have a complete, private preference list and a unit demand over the items. Our focus is on allocation mechanisms that do not involve any monetary payments. We consider two natural measures of social welfare: the ordinal welfare factor which measures the number of agents that are at least as happy as in some unknown, arbitrary benchmark allocation, and the linear welfare factor which assumes an agent's utility linearly decreases down his preference lists, and measures the total utility to that achieved by an optimal allocation. We analyze two matching mechanisms which have been extensively studied by economists. The first mechanism is the random serial dictatorship (RSD) where agents are ordered in accordance with a randomly chosen permutation, and are successively allocated their best choice among the unallocated items. The second mechanism is the probabilistic serial (PS) mechanism of Bogomolnaia and Moulin [8], which computes a fractional allocation that can be expressed as a convex combination of integral allocations. The welfare factor of a mechanism is the infimum over all instances. For RSD, we show that the ordinal welfare factor is asymptotically 1/2, while the linear welfare factor lies in the interval [.526, 2/3]. For PS, we show that the ordinal welfare factor is also 1/2 while the linear welfare factor is roughly 2/3. To our knowledge, these results are the first non-trivial performance guarantees for these natural mechanisms

On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{\Omega\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.Comment: 21 pages, Conference version to appear in ESA 201

Emission Spectra from Internal Shocks in Gamma-Ray-Burst Sources

Unsteady activity of gamma-ray burst sources leads to internal shocks in their emergent relativistic wind. We study the emission spectra from such shocks, assuming that they produce a power-law distribution of relativistic electrons and posses strong magnetic fields. The synchrotron radiation emitted by the accelerated electrons is Compton up-scattered multiple times by the same electrons. A substantial component of the scattered photons acquires high energies and produces e+e- pairs. The pairs transfer back their kinetic energy to the radiation through Compton scattering. The generic spectral signature from pair creation and multiple Compton scattering is highly sensitive to the radius at which the shock dissipation takes place and to the Lorentz factor of the wind. The entire emission spectrum extends over a wide range of photon energies, from the optical regime up to TeV energies. For reasonable values of the wind parameters, the calculated spectrum is found to be in good agreement with the burst spectra observed by BATSE.Comment: 12 pages, latex, 2 figures, submitted to ApJ

Dairy farmer and farm staff attitudes and perceptions regarding daily milk allowance to calves

The benefits of feeding calves more milk are increasingly being recognized by dairy farmers. However, most producers have still not implemented higher feeding plans. The aim of the present study was to gain a deeper understanding of farmer and farm staff attitudes, and the perceptions and factors considered in their decision-making regarding daily milk allowances. We collected data through focus group interviews with dairy farm-ers, farm managers, and calf-care workers who were selected using purposive and snowball sampling. In total, 40 persons (24 women and 16 men) joined a focus group interview (6 in all, each with 5-8 participants). Interviews were recorded, and recordings were tran-scribed and analyzed thematically. Participants had contrasting opinions about the minimum, maximum, and recommended daily milk allowances to their calves. Their suggested lowest daily milk allowance to sustain animal welfare ranged from 4 to 8-10 L and the maxi-mum allowance from 6 to 15 L. We found that farmers' and farm staff's choices and recommendations of milk-feeding protocols were influenced by a large number of factors that could be grouped into 4 themes: (1) Life beyond work, (2) Farm facilities and equipment, (3) Care of the calves, and (4) Profitability and pro-duction. Participants' considerations were similar and aimed to maximize daily milk allowance based on farm conditions. However, the allowances they described as optimal for their calves often differed from what they considered practically feasible. We found that the care of the calves and the well-being of the owners and the staff was central in the participants' decision-making, but that this care perspective was challenged by the social and economic sustainability of the farm. Most participants fed their calves twice daily and did not think that increasing that number would be practically feasible. Our results indicate that the participants' viewpoints regarding calves were important for their decision-making about milk allowances. We suggest that a more holistic perspective should be used when advising farmers about milk allowances, putting particular emphasis on the caring and social sustainability aspects of the individual farm